Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Dyson–Schwinger Equationsin Euclidean and Minkowski space

Pieter Maris

Dept. of Physics and AstronomyIowa State University

Ames, IA 50011

and

Departamento de FısicaInstituto Tecnologico de Aeronautica

Sao Jose dos Campos, Brazil

pmaris@iastate.edu

XXXIX Brazilian Meeting on Particles and Fields,Campos do Jordao SP, Sept 2018

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 1 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Outline

Motivation: Hadron Physics

Nonperturbative QCD

Dyson–Schwinger Equations

Mesons as qq bound states

Meson form factors

DSEs in Minkowski space

Concluding remarks

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 2 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Hadron physics

Hadrons as bound states of quarks and gluons

I Nucleons as bound states of three valence-quarks

I Mesons as bound states of a quark and an antiquark

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 3 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Mass spectrumHadron physics – mass spectrum

Where do hadronsget there mass fromif the quark massesare only a few MeV?

PDG: at scale µ = 2 GeV2 MeV < mu,d < 8 MeV

0 -

1 -

1/2 +

3/2 +0.0

0.5

1.0

1.5

hadr

on m

ass

[G

eV]

Why are pions so much lighter than all other hadrons ?

hyperfine splitting??

Delta ↑↑↑ 1.23 GeV

proton ↑↓↑ 0.94 GeV

rho ↑↑ 0.77 GeV

pi ↑↓ 0.14 GeV

Hadron Physics based on the Equations of Motion of QCD – p.3/37P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 4 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Scattering observables

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 5 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Quantum Chromo DynamicsI Relativistic Quantum Field Theory

L(ψ, ψ,A) = ψ

(iγµ(∂µ + ig λ

2 Aµ

)−m

)ψ − 1

4FµνFµν + gauge fixing

I Dynamical chiral symmetry breakingI Quarks and gluons are confined

I they have never been observed in isolationI only colorless bound states of quarks and gluons are observed

Confinement

Experimentally, only color-singlet hadrons are observedNo free quark has ever been observed

Conjecture: Confinement

Only color-singlet combinationsof the elementary fields can be observed

mesons baryons

Introduction to Hadron Physics – p.15/18

I Need nonperturbative methods

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 6 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Nonperturbative methods

I Lattice QCD

I Dyson–Schwinger Equations

I Light-Front QCD

I Stochastic quantization

I AdS/QCD holography

I . . .

= +p -1 -1p

k

p-kDirac’s forms of relativistic dynamics [Dirac, Rev.Mod.Phys. ’49]

In relativity, t = x0 is not the only choice of “time”, which dictates thedirection of the dynamical evolution.

instant form front form point form

t = x0 x+ , x0 + x3 τ ,√t2 − ~x2 − a2

H = P 0 P− , P 0 − P 3 Pµ

~P , ~J ~P⊥, P+, ~E⊥, E+, Jz ~J, ~K

~K,P 0 ~F⊥, P− ~P , P 0

p0 =√~p2 +m2 p− = (~p2

⊥ +m2)/p+ pµ = mvµ (v2 = 1)

P± , P 0 ± P 3, ~P⊥ , (P 1, P 2), x± , x0 ± x3, ~x⊥ , (x1, x2), Ei = M+i,E+ = M+−, F i = M−i, Ki = M0i, J i = 1

2εijkM jk.

7/55

time variable

quantizationsurface

Hamiltonian

kinematical

dynamical

dispersionrelation

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 7 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Nonperturbative QCD: Lattice

Simulate QCD action on high-performance supercomputersI Discretize Euclidean space-timeI Use Monte Carlo methods to sample path integralsI In principle exactI In practice:

I finite lattice size effects L = N × a <∞I finite lattice spacing effects a > 0

I Additional problem:I light quarks are numerically expensive

I Need extrapolation methodsI continuum limit L→∞, a→ 0I chiral limit, realistic u and d quark masses

for review, see e.g. Fodor, Hoelbling, Rev. Mod. Phys. 84, 449 (2012)

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 8 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Select lattice QCD highlights

I Mass spectrumScience 322, (2008)

I Mass splittingsScience 347, (2015)

I Nucleon axial couplingNature 558, 91 (2018)

Fig. from Durr et al. Science 322, 1224 (2008)

0

500

1000

1500

2000

M[M

eV]

π

K

ρK* N

ΛΣ

ΞΔ

Σ*Ξ*

Ω

experimentwidthinputprediction

however, challenging issues remain, despite its succcesses

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 9 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Challenging problems in lattice QCD

I Form factors can only be evaluated at limited discrete values of Q2

I Lattice QCD is based on Euclidean space-time (imaginary time)I Applications that suffer from the sign problem

I real time observables, typically defined on the light-front: e.g.distribution amplitude (DA), generalized parton distribution (GPD)

I finite density simulations

I Possible solutionsI calculate moments of DAs, GPDs

I limited to lowest couple of moments, which is insufficient to reliablyreconstruct DAs, GPDs, etc.

I implicitly relies on conventional analytic structure of Green’s functionsI deform ’the domain of the path integral to a manifold in complex

field space that maximizes the average sign’Alexandru, Bedaque, Lamm, Lawrence, PRD 97, 094510 (2018)

I stochastic quantization (yesterday, Krein)I . . .

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 10 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Nonperturbative QCD: Dyson–Schwinger Equations

-1=

-1- 1/2 - 1/2

- 1/2 - 1/6

+ +

-1=

-1−

-1=

-1−

I Infinite hierarchyof coupled integral eqnsfor Green’s functions of QCD

I Reduce to pQCDin weak coupling limit

I Nonperturbative

I Truncation neededI Constraints on truncation

I Preserve symmetriesI Self-consistency

I aka Dyson–Schwinger Eqns(yesterday, Aguilar)

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 11 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Quark propagator in Euclidean metric

S0(p) =1

i 6p + mq−→ S(p) =

Z (p2)

i 6p + M(p2)=

1i 6pA(p2) + B(p2)

I One-loop perturbative correction to mq

M(p2) = mq

(1− 3α

4πln

[p2

m2q

]+O(α2)

)

I Perturbatively, running quark mass function M(p2)proportional to current quark mass mq(µ)

M(p2)large p2

−→ mq(µ)

(ln(p/ΛQCD))γm

with anomalous dimension γm = 12/(2Nc − 2Nf )

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 12 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Nonperturbative quark propagator

S0(p) =1

i 6p + mq−→ S(p) =

Z (p2)

i 6p + M(p2)

I Satisfies Dyson–Schwinger equation

= +p -1 -1p

k

p-k

S(p)−1 = i 6p Z2 + mq(µ) Z4 + Z g1

∫d4k

(2π)4 g2Dµν(q)γµλa

2 S(k)Γν(k ,p)λa

2

I Nonlinear integral equation for the quark propagatorI coupled nonlinear integral equations for M(p2) and Z (p2)

I Allows for a nontrivial solution M(p2) 6= 0 even if mq = 0 providedthe long-range part of the interaction is sufficiently strong

I dynamical chiral symmetry breakingI pions are the (near) massless Goldstone bosons

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 13 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Rainbow truncation of the quark DSE

I Rainbow truncation for quark DSEZ g

1 g2Dµν(q) Γν(k ,p) −→ 4παmodel(q2) Dfreeµν (q) γν

Nonperturbatively dressed quark propagator

Rainbow truncation for quark DSEZg1 g

2Dµν(q) Γν(k, p) −→ 4παmodel(q2)Dfree

µν (q) γν

= + + + . . .

+ + + . . .

+ + . . .

Hadron Physics based on the Equations of Motion of QCD – p.11/37P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 14 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Model for effective interaction

I Rainbow truncation for quark DSEZ g

1 g2Dµν(q) Γν(k ,p) −→ 4παmodel(q2) Dfreeµν (q) γν

I Assume dressed vertex times dressed gluon has the sametensor structure as bare vertex times free gluon propagator

I Use Landau gaugeI in principle we could

use any covariant gauge

I Use pQCD for UV behaviorI Model for IR behavior of α(q2)

fitted to give chiral condensate〈qq〉 = −(240 MeV)3

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 15 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Model results for nonperturbative quark propagator

I Rainbow truncation for quark DSEZ g

1 g2Dµν(q) Γν(k ,p) −→ 4παmodel(q2) Dfreeµν (q) γν

I Evolution from constituent quark mass to current quark massI M(p2) connects constituent mass with perturbative QCD

10-2

10-1

100

101

102

103

104

p2 [GeV

2]

10-5

10-4

10-3

10-2

10-1

100

101

M(p

2)

[G

eV

]

b-quark

c-quark

s-quark

u/d-quark

chiral limit

Fig. from PM, Roberts,PRC56, 3369 (1997)

Nonzero mq(µ)

M(p2)large p2

−→ mq(µ)

(ln(p/ΛQCD))γm

Dynamical χSB

M(p2)large p2

∝ 〈qq〉0p2 (ln(p/ΛQCD))1−γm

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 16 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Nonperturbatively dressed quark propagator

I Predictions from solution of the quark DSEhave been confirmed by lattice simulations of QCD

I Detailed comparison lattice simulations and DSE soln possiblePM, Raya, Roberts, & Schmidt, EPJA18, 231 (2003)

0 1 2 3 4

p [GeV]

0

0.1

0.2

0.3

0.4

0.5

M (

p)

[G

eV]

0 1 2 3 4

p [GeV]

0

0.2

0.4

0.6

0.8

1

Z(p

)

Lattice-inspired DSE model: Bhagwat, Pichowsky, Roberts, Tandy, PRC68, 015203 (2003)Quenched lattice data: Bowman, Heller, Leinweber, Williams, NP Proc.Suppl.119, 323 (2003)

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 17 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Hadrons

I Bound states of nonperturbatively dressed quarksI Pole in color-singlet n-point functions of QCDI Bound state amplitudes Γ describe coupling between

I meson and quark-antiquark pair

G(4) ∼ Γ(p1,p2; P)Γ(k1, k2; P)

P2 + M2meson

I baryon and three quarks

G(6) ∼ Γ(p1,p2,p3; P)Γ(k1, k2, k3; P)

P2 + M2baryon

I Euclidean metric: mass poles at P2 = −M2hadron

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 18 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Mesons

Quark-antiquark bound states satisfy homogeneousBethe–Salpeter equation at mass pole P2 = −M2

meson

ΓH(p; P) =

∫d4k

(2π)4 K (p, k ; P) S(k + P/2) ΓH(k ; P) S(k − P/2)

=

P p + P/2

p - P/2

P p + P/2

p - P/2

K

I K (p, k ; P): amputated qq scattering kernelI Quark propagators nonperturbatively dressed

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 19 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Euclidean formulation

I Meson BSA functions of two independent variables: p2 and p · PI Rest-frame P = (iM,0,0,0)I Relative momentum p Euclidean

I p2 space-likeI p · P imaginary in rest-frame

I Integration variable k Euclidean

I Quark propagator argumentsk2 ± k · P + M2/4 become complex

I Constituent propagators:no problem for bound states,that is, for M < 2m

I e.g. constituent quark massof 400 MeV fine for π and ρ

-0.2 0 0.2 0.4 0.6 0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6 M2 = 1.2 GeV

2

M2 = 0.8 GeV

2

M2 = 0.4 GeV

2

M2 = 0.2 GeV

2

M2 = 0

m2 = 0.16 GeV

2

complex momentum plane

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 20 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Analytic continuation of dressed quark propagator

A(p2) = 1 +

∫d4k4π3

α(q2)

q2A(k2) K A(p2, k2,p · k)

k2 A2(k2) + B2(k2)

B(p2) = mq(µ) +

∫d4k4π3

α(q2)

q24 B(k2)

k2 A2(k2) + B2(k2)

I Fit Euclidean solution with your favorite functionI results will (strongly) depend on choice of functional form

I Use Taylor expansion of Euclidean solutionI limited range, but should be okay for light systems

I Calculate A(p2) and B(p2) at complex momenta p2

after solving quark DSE on real Euclidean axisI only correct if effective interaction α vanishes at q2 = 0, otherwise,

pinch-singularity forces integration path dk through k = pI Analytic continuation of quark DSE into complex plane

I can be done, but is nontrivial

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 21 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Analytic structure of rainbow DSE solution

I Landau gauge, bare vertex, pQCD for UV behavior of couplingI Model for IR behavior of α(q2) fitted to give chiral condensate〈qq〉 = −(240 MeV)3

I Solution appears to have pair ofcomplex-conjugate singularitiesrather than real mass-like pole

I Allows for bound state calculationsof light mesons up to massesof about ∼ 1.2 GeV2

I Similar structure foundfor a wide range of models,as well as from lattice QCD

I Indication of confinement ?-0.2 0 0.2 0.4 0.6 0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6 M2 = 1.2 GeV

2

M2 = 0.8 GeV

2

M2 = 0.4 GeV

2

M2 = 0.2 GeV

2

M2 = 0

complex conj.singularities in S

complex momentum plane

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 22 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Rainbow–Ladder truncation

I Rainbow truncation for quark DSEZ g

1 g2Dµν(q) Γν(k ,p) −→ 4παmodel(q2) Dfreeµν (q) γν

I Ladder truncation for Bethe–Salpeter EqnK(p, k ; P) −→ −4παmodel(q2) Dfree

µν (q) γµλa

2 ⊗ γν λa

2with q = p − k

K

I First term in systematic expansion of the kernelI Corrections are small in pseudoscalar and vector channels

Bender, Roberts, & Smekal, PLB380, 7 (1996)I Respects Axial-Vector WTI

−iPµ Γ5µ(k ; P) = S−1(k+) γ5 + γ5 S−1(k−)− 2 mq(µ) Γ5(k ; P)

−→ pions as (massless) Goldstone bosonsP. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 23 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Results for Maris–Tandy model interaction

0 0.2 0.4 0.6 0.8 1

mq(19 GeV) [GeV]

0

1

2

3

meson m

ass [G

eV

]

expt. dataqq V mesonqq PS mesonsq PS mesonuq PS meson

K

φ

ρ

π

D

Ds

ηc

J/ψ

Fitted to π and K mesons,applied to ρ and K ∗ inPM, Tandy, PRC60,055214 (1999)

Fig. from PM, Tandy,nucl-th/0511017

I ρ-π mass splitting driven by chiral symmetry breaking,not by fine-tuning the hyperfine splitting

I Extension to heavier mesonsusing same interaction for light quarks and for heavy quarksP. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 24 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Frame independence

I In principle, all frames are equivalentI In practice, some frames are easier than others . . .I Meson BSE: discrete solutions at P2 = −M2

ΓH(p; P) =−43

∫d4k

(2π)4 4π α((p − k)2)Dµν(p − k) γµ S(k + P/2)ΓH(k ; P)S(k − P/2) γν

I rest frame: P = (iM,0,0,0)I moving meson: P = (iE ,q,0,0) with E2 = M2 + q2

0 0.2 0.4 0.6 0.8 1

meson 3-momentum q [GeV]

0

0.2

0.4

0.6

0.8

mas

s, d

ecay

const

ant

[G

eV]

rho massrho decay constant

pion mass

pion decay constant

I Numerically more expensiveI more independent variablesI p · P complex,

instead of imaginaryI Limited by analytic structure

of quark propagators

PM, Tandy, nucl-th/0511017

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 25 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Meson electromagnetic form factors

π

π

γ

I Quark propagatorI Meson BS amplitude

I in moving frameI Quark-photon coupling

Nonperturbative QFT approachbased on QCD dynamicsreproduces pQCD resultsexplicitly relativistic, Poincare invariant

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 26 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Quark-photon coupling

I Electromagnetic current conservation ∂µJµ = 0I Vector Ward–Takahashi identity

i Pµ Γµ(k+, k−; P) = S−1(k + P/2)− S−1(k − P/2)

I Inhomogeneous Bethe–Salpeter equationfor the quark-photon vertex

I Same kernel K as meson bound state eqnI Solve for

ΓTµ (k+, k−; P) =

8∑i=0

T iµ(k ,P)Fi(k2, k · P; P2)

I Guarantees electromagnetic current conservationP. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 27 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Pion electromagnetic form factor

Λµ(P,Q) = 2 Pµ Fπ(Q2) = Nc

∫d4k

(2π)4 Tr[Γπ S iΓµ S Γπ S

]I Need at least one BS amplitude in moving frameI Within numerical accuracy,

results independent ofI choice integration variablesI form factor frame P-Q/2 P+Q/2

Q

k-P/2

k+P/2-Q/2 k+P/2+Q/2

I Note: Form factor has pole at vector meson massesI quark-photon vertex BSE has poles at Q2 = −Mρ,ω,φ,...

Γµ(k ; Q) ' fρ Mρ

Q2 + M2ρ

Γρµ(k ; Q)

P k + P/2

k - P/2

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 28 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Pion electromagnetic form factor

-0.2 0.0 0.2 0.4 0.6 0.8

Q2 [GeV

2]

0.0

1.0

2.0

3.0

4.0

Fπ(Q

2)

0.0 0.1 0.2

0.7

0.8

0.9

1.0

1.1

BSE solution

bare vertex

PM and Tandy, PRC62,055204 (2000) [nucl-th/0005015]

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 29 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Pion electromagnetic form factor

0

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2 2.5 3 3.5 4

Q2 (GeV/c)

2

Q2 F

π (

Ge

V/c

)2

QCD Sum Rules (Nesterenko, 1982)

pQCD (Bakulev et al, 2004)

BSE-DSE (Maris and Tandy, 2000)

Disp. Rel. (Geshkenbein, 2000)

CERN π-e scattering

DESY (Ackermann)

DESY (Brauel)

JLab (Tadevosyan)

this work

Tadevosyan et al. [Fpi2 Collaboration], nucl-ex/0607007; Horn et al. [Fpi2 Collaboration], nucl-ex/0607005

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 30 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Euclidean vs. Minkowski space

I Both lattice QCD and DSEsconventionally formulated in Euclidean space

I Singularities in the nonperturbative quark propagator limit theanalytic continuation of the propagator into the complex plane

I limits the calculation of pion form factors to Q2 < 4 GeV2

I limits elastic form factor calculations of heavier mesonsand baryons to small range of Q2

I Advantages of Minkowski space formulationI no need for BS amplitude in moving frameI asymptotic form factorsI obtain observables typically defined on the light-front, such as

distribution amplitude (DA), generalized parton distribution (GPD)I access to light-front wavefunctions

I So, why not perform calculations directly in Minkowski spaceI lattice QCD: sign problemI DSEs: why not?

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 31 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Challenge: Minkowski space calculationsin collaboration with Frederico, de Paula, Ydrefors, Duarte, Castro

I Solve DSE and/or BSE directly in Minkowski spaceProblem: how to deal numerically with mass-like poles

I fermion DSE: expect mass-like singularities in timelike region, butlocation is unknown until a self-consistent solution has been found

I bound state BSE: if propagator singularities in timelike region areknown, an explicit iε prescription should work . . .

I confinement: not clear whether or not to expect mass-likepropagator singularities in timelike region

I Solve DSE and/or BSE ’near’ Minkowski spaceI explicit Wick rotation back to Minkowski space, starting

from converged solution in Euclidean space work in progress

I Nakanishi formulationI can be implemented for BSE, using constituent propagatorsI implementation for fermion DSE more difficult work in progress

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 32 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Un-Wick-rotating the fermion DSE to Minkowski space

I Fermion DSE in rainbow truncation

A(p2) = 1 + g2∫

d4k(2π)4

A(k2) K A(p2, k2, p · k ; ξ)

k2 A2(k2) + B2(k2)

B(p2) = m0 + g2∫

d4k(2π)4

B(k2) K B(p2, k2, p · k ; ξ)

k2 A2(k2) + B2(k2)

I Analytic continuation of Euclidean quark DSEI choose p0 axis and write four-vectors as p = (p0, ~p)I replace 4-dim Euclidean integral by k0 integral and 3-dim integral

A(p0, ~p2) = 1 + g2∫ ∞

−∞dk0

∫d3k

(2π)4

A(k0, ~k2) K A(p0, k0, ~p, ~k , ~p · ~k)

(k20 + ~k2) A2((k2

0 + ~k2)) + B2((k20 + ~k2))

B(p0, ~p2) = m0 + g2∫ ∞

−∞dk0

∫d3k

(2π)4

B(k0, ~k2) K B(p0, k0, ~p, ~k , ~p · ~k)

(k20 + ~k2) A2((k2

0 + ~k2)) + B2((k20 + ~k2))

I Wick-rotate p0 and k0 from Euclidean back to Minkowski space

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 33 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Un-Wick-rotating the fermion DSE

I Wick-rotate p0 and k0 from Euclidean back to Minkowski spacein small increments δ p0 → exp(−iδ) p0

k0 → exp(−iδ) k0

I Solve DSE iteratively along rotated p0 axis,starting with solution at previous value of δ

I Use Pauli–Villars regulator to remove UV divergences

0.1 1 10 100k

0

0

0.2

0.4

0.6

0.8

1

A(p

0)

Euclideanδ = π/16δ = 2π/16δ = 3π/16δ = π/ 4δ = 5π/16δ = 6π/16

m = 0.5, µ = 0.1, Λ = 10, α = 1.0

0.1 1 10 100p

0

0

0.4

0.8

1.2

1.6

2

B(p

0)

Euclideanδ = π/16δ = 2π/16δ = 3π/16δ = π/ 4δ = 5π/16δ = 6π/16

m = 0.5, µ = 0.1, Λ = 10, α = 1.0

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 34 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Revealing mass-like pole at weak coupling

I Space-like region given by k0 = 0I Approximate time-like region by ~k = 0, even if δ < π/2I Look at propagator, rather than inverse propagatorI Suggest mass-like pole in time-like region

I need to improve numerics for strong-coupling case,even though A and B looked smooth and converged

-2 -1 0 1 2 3 4 5k

2

-2

-1

0

1

2

3

4

5

B /

(A

2 k

2 -

B2)

δ = 10π/32δ = 11π/32δ = 12π/32δ = 13π/32δ = 14π/32

m = 0.5, µ = 0.1, Λ = 10, α = 0.1

Time-like

Space-like

-4 -2 0 2 4 6 8 10 12 14k

2

-3

-2

-1

0

1

2

3

4

B /

(A

2 k

2 -

B2)

δ = 10π/32δ = 11π/32δ = 12π/32δ = 13π/32

m = 0.5, µ = 0.1, Λ = 10, α = 1.0

Time-like

Space-like

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 35 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Thoughts on DSE in Minkowski space

I Wick-rotating back from Euclidean space to Minkowski spaceseems to work, at least for weak coupling

I can get close to Minkowski axis for weak coupling in Landau gaugeI need to extract pole mass and compare with Schwinger function

Alkofer, Detmold, Fischer, PM, PRD70, 014014 (2004)I need to include nonperturbative vertex Ansatz

e.g. Ball–Chiu vertex, Curtis–Pennington vertex, . . .I Analytic structure for confined particles in QCD is unclear

I lattice QCD and most strong-coupling DSEs suggest pair ofcomplex-conjugate singularities

I suitable model for running coupling in time-like region?I If quark and gluon propagator have pair of complex-conjugate

singularities, then how do we properly connect Minkowski space(observables, light-front wavefunctions) with nonperturbativecalculations based on Euclidean formulation?

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 36 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Bound state BS amplitude in Minkowski space

Γ(k ; p) = g2∫ ∞

−∞dq0

∫d3q

(2π)4 K (k , q; p) S(q + p/2) Γ(q; P) S(q − p/2)

I Domain of k2 and k · p for two-body BSA is frame-independentI rest-frame: p = (M,0)I moving meson: p = (p0, ~p) with p2

0 = M2 + ~p2

-12 -10 -8 -6 -4 -2 0 2 4 6 8

k2 = k

0

2 - k

v

2

-6

-4

-2

0

2

4

6

k.p

( =

k0 M

in r

estf

ram

e)

kinematic regionnot accessed in BSA

location of poles in constituent propagators

at k.p = ±(k2 - m

2 + M

2/4)

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 37 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Un-Wick-rotating scalar BSE to Minkowski space

I For simplicity, consider scalar two-body bound statewith massive or (near) massless one-boson exchange kernel

I Solve for Γ(k ; p) as function of k0 and ~k for decreasing angles θstarting from θ = π/2 (Euclidean)

I Singularies develop for k0 = |~k |+ (m + µ−M/2)as one approaches Minkowski axis

-4 -3 -2 -1 0 1 2 3 4

k0/m

-40

-20

0

20

40

60

80

Re[Γ

(k0,

kv)]

kv/m = 0.067

kv/m = 0.33

kv/m = 1.0

kv/m = 3.0

α = 5.48, µ/m = 0.2, M/m = 1.0, θ = π / 32

-4 -3 -2 -1 0 1 2 3 4

k0/m

-20

0

20

40

60

80

100

Im[Γ

(k0, k

v)]

kv/m = 0.067

kv/m = 0.33

kv/m = 1.0

kv/m = 3.0

α = 5.48, µ/m = 0.2, M/m = 1.0, θ = π / 32

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 38 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Nakanishi integral representationNakanishi, Phys.Rev. 130, 1230 (1963); Prog.Theor.Phys.Suppl. 43, 1 (1969)

I For propagators (2-point functions) of asymptotic states

S(p) = −i∫ ∞

0dγ

g(γ)

(γ + m2 − k2 − iε)n

n = 1 gives usual Kallen–Lehmann representationI For two-body BSA of constituents with asymptotic states

Γ(k ; p) = −i∫ 1

−1dz∫ ∞

0dγ

g(γ, z)

(γ + m2 −M2/4− k2 − k · p z − iε)n

I Used forI 2-body scalar BSE Kusaka, Simpson, Williams, PRD56, 5071 (1997)I fermion DSE and BSE Sauli, JHEP 0303, 1 (2003)

I Applicability to confined states unclear

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 39 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Comparison with Nakanishi representation

Solve BSE using Nakanishi representation for χ(k ; p)

χ(k ; p) ≡ ∆(k + p/2) Γ(k ; P) ∆(k − p/2)

= −i∫ 1

−1dz∫ ∞

0dγ

g(γ, z)

(γ + m2 −M2/4− k2 − k · p z − iε)3

-4 -3 -2 -1 0 1 2 3 4

k0/m

0

Γ(k

0, k

v)

Re[Γ] (Wick rot)

Im[Γ] (Wick rot)

Re[Γ] (Nakanishi)

Im[Γ] (Nakanishi)

α = 5.48, µ/m = 0.2, M/m = 1.0, θ = π/16, kv/m = 0.067

I Note: χ(k ; P) containsconstituent poles atk ·p = ±(k2−m2+M2/4),as well as singularitiesat k0 = (m + µ−M/2)

I Calculate Γ using∆−1 χ ∆−1

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 40 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Perspectives on BSE in Minkowski space

I Wick-rotating back from Euclidean space to Minkowski spaceseems to work, at least for scalar two-body bound states

I can get close to Minkowski axisI BSA numerically more stable than with Nakanishi representation

I Questions (hopefully addressed soon)I can we extract observables from calculations

close to the Minkowski axis?I can we calculate the elastic form factor with θ ≈ 0?I can we extract the light-front wavefunction

I Nakanishi representation enables calculation of elastic formfactors over entire space-like region, including the asymptotics

I Nakanishi representation can be extended and modified to usepropagators with a pair of complex-conjugate singularities

I interpretation of complex-conjugate singularities?

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 41 / 42

Hadron Physics QCD DSEs Mesons Form factors Minkowski Conclusion

Concluding remarks

I Nonperturbatively dressed quark propagator connectscurrent quark mass and pQCD with constituent quark models

I Mesons can be described accurately as qq bound statesI challenge: ’observables’ defined on the light-front (GPDs, . . . )

I Preserve relevant symmetries and propertiesI Poincare invarianceI axial-vector WTI for chiral symmetry & chiral symmetry breakingI vector WTI for electromagnetic observables

I Analytic structure of Green’s functions ?I Nakanishi representation work in progressI explicit (backward) Wick rotation to Minkowski space

work in progressI Baryons as 3-quark bound states is possible,

thanks to increased computational resourcesI Beyond rainbow-ladder truncation

P. Maris (ISU & ITA) DSEs XXXIX ENFPC, Sept 2018 42 / 42